Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details).

IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has past, I've spent days going through the results, and I have collated them all here: Errata for Cassels-Froehlich. I will update this file as comments come in. The London Mathematical Society would like to know all the errors I've made myself, by 12th of February, so feel free to let me know of anything, however trivial!

EDIT: file updated 15th Feb.

Thanks to everyone who helped.

This one is very borderline and I certainly won't be offended/surprised if it gets closed. [EDIT: I clearly misjudged this---the question has a good few upvotes now.]

The London Maths Society (LMS) are thinking of (indeed, actively pursuing the idea of) reprinting "Algebraic number theory" Ed. Cassels and Froehlich. Hence the LMS had to go about contacting the authors of the original articles. When they contacted Serre he replied "sure reprint my articles, but please include the erratum that I indicated in my completed works." Sure enough, he had found a slip in one of his articles (in the statement of the "ugly lemma"---Serre went on to say that this had taught him not to abuse lemmas, as they might bite back!) and had taken the trouble to fix it when his completed works were published.

I looked over the thread from last October about the errata database but the database doesn't seem to contain this book. On the other hand it is surely a very widely-read book. I think I just found a typo in the definition of a co-induced module on p98: I think the $G$-action on $Hom_{G'}(\mathbf{Z}[G],A')$ ($G'$ a subgroup of $G$ and $A'$ a $G'$-module) should be that $g\in G$ sends $\phi$ to the function sending $g'$ to $\phi(g'g)$, not what the authors say (what they say isn't even an action as far as I can see, unless I made a slip).
The notation is also terrible: $g'$ is in $G$, it seems to me.

Does anyone else have any scrawled marginal notes in their copies of Cassels-Froehlich about typos or other things that the LMS can fix? They are planning on having an erratum page at the beginning of the book when they reprint it.

EDIT: someone from the LMS got in touch with me to say that the only errata they would dare publish would be errata that had been confirmed by the authors, or someone "of a similar standing", so in fact it might be the case that not everything mentioned here gets put in the LMS erratum.

If this thread does get closed, feel free to email comments to buzzard at imperial.ac.uk.

Edit: the LMS have set a deadline of 1st Feb. After this point I (KB) will put all the errata we have caught into one file and the LMS will send it to the authors, asking for their approval.

Edit: Anton and Ilya have suggested that really this would be better if it had one big answer rather than lots of smaller ones. But let me persist with the "lots of smaller ones" for the time being, because I am still getting emails with non-trivial lists in from different sources and, although I want to put everything together into one pdf file, I don't really want to do it until I am pretty sure no more is coming in. On the other hand these partial lists have definitely been of help to some people, e.g. I've had emails saying "I have a big list of corrections; here are the ones that haven't already been mentioned."

Edit: OK, so all the people whom I was almost sure would have comments have now got back to me. I posted everything in one "burst" so as to only bump this post to the top one last time. What I will do now is to compile everything I have now (on the basis that I am not expecting much more) into one pdf list, and post a link to it.

$\begingroup$Why is this borderline? This actually seems to me an ideal use of the collected attention of Math Overflow to do something that helps the community more generally.$\endgroup$
– JSEJan 11 '10 at 16:20

3

$\begingroup$I agree, this seems like an excellent question. And I'm glad to hear that Cassels and Froelich may be reprinted!$\endgroup$
– David E SpeyerJan 11 '10 at 16:22

3

$\begingroup$Hmm. It's "borderline when it comes to what I want MO to be" ;-) (which is just loads of fun precisely-worded questions with precise answers). But judging by the upvotes I have misjudged this. I also emailed some people asking them if they had seen any typos. I'll post anything I get.$\endgroup$
– Kevin BuzzardJan 11 '10 at 16:24

6

$\begingroup$In my opinion this is an excellent example of a good community wiki question!$\endgroup$
– GMRAJan 11 '10 at 16:32

4

$\begingroup$I thought of a use of upvotes/downvotes. If you think the typos really are typos, vote up :-) If you think it's reader error vote down!$\endgroup$
– Kevin BuzzardJan 11 '10 at 22:06

And here's one which I spotted: I think that the last
full sentence at the bottom of p98 is wrong. I think the "action" they
define is not an action, and I think the first couple of sentences of section 4
should be:

Let $G'$ be a subgroup of $G$. If $A'$ is a $G'$-module, we can form the $G$-module
$A=Hom_{G'}(\Lambda,A')$. We give $A$ the following $G$-module structure: if
$\phi\in A$,then $g.\phi$ is the homomorphism $h\mapsto\phi(hg)$. Then we have...

$\begingroup$Bill Stein independently got in touch with me to tell me about the two typos on p99 (he's giving a talk on that chapter in 30 minutes' time!)$\endgroup$
– Kevin BuzzardJan 11 '10 at 23:10

3

$\begingroup$I remember Serre explaining long ago that the reference to Stalin was a mischievous joke linked to NATO's sponsorship of the conference.$\endgroup$
– Georges ElencwajgJan 12 '10 at 12:48

$\begingroup$On p.140 Prop 1. line 7 of the proof, it is very confusing to write $\delta(r/n)\in\hat{H}^0(G,\mathbb{Z})$ and $\delta(r/n) =r$, since we know $\hat{H}^0(G,\mathbb{Z}) =\mathbb{Z}/n\mathbb{Z}$. I think this is a typo? Actually, I think the entire proof of Prop.1 is just restating the fact of line 6. It is tautological.$\endgroup$
– Ying ZhangJan 10 '11 at 1:13

Dominique Bernardi points out that the formula for $\phi_a$ is wrong on line 1 of p96. This is a delicate one. The issue is what the definition of the action of $G$ on $A*$ is (NB that starshould be a superscript but the TeX interpreter is playing up on me). It is not explicitly defined in the paper, but the authors claim that $A*$ is co-induced on p96 and I think that this is hence an implicit definition. I figure that probably $A*=Hom(\Lambda,A)$ with the $G$-action on $A$ being thought of as trivial, so for $f\in A*$ we have $(\gamma.f)(\lambda)=f(\gamma^{-1}\lambda)$. If I got it right then the correct definition of $\phi_a$ is $\phi_a(g)=g^{-1}a$.

$\begingroup$Oesterle points out that the "standard" definition of a co-induced module is Hom(Lambda,X) with G-action (g.f)(lambda)=f(lambda.g). If this definition is used then what the authors write seems to be OK. However it seems to me that implicit in the article is a "non-standard" (but isomorphic to the standard) definition.$\endgroup$
– Kevin BuzzardJan 12 '10 at 13:57

This answer is just to bump this post up to the front page for the final time. I typed up all the errata I heard into one pdf file and put it here. The London Maths Society would like comments, if any, by Friday 12th Feb.

Eric Bach writes to tell me that there should arguably be a +- sign in front of the formula for the discriminant of the $m$th cyclotomic field appearing on pp88--89 (or perhaps it should be flagged in some way that this is what the the "classical" discriminant is).

Here's one I didn't see on the list on mathoverflow: In exercise 2,
part 10, equation (**) (page 353, line 4) should be

$$\prs{\lambda}{b}=\prod_{v\in S}(b,\lambda)_{v}$$

not

$$\prs{\lambda}{b}=\prod_{v\in S}(\lambda,b)_{v}$$

(\prs = power residue symbol). The right-hand side should be the
reciprocal of what it is in the book.

I recall also having trouble getting the signs in part 14 of that
exercise (on cubic reciprocity) to work out, perhaps because of the mess
of algebra, perhaps because some of the computations depend on part
10. I'm suspicious of equation (**); I'll try to check it again.

Below my 51 errata that I didn't see on your list or in William Stein's mail
yet. Most are of a typographical nature, but some have mathematical
substance. I did at the present occasion not verify the correctness of those.

And: I did not do any proofreading of my list either!! I trust you will apply
your own sound judgment.

Good luck!!

And best regards,

Hendrik

Errata for Cassels-Fr\"ohlich
copied by Hendrik Lenstra from his own copy
Jan 13, 2010

Page 3, Proposition 1. This Proposition is misstated, and the proof has the
wrong reference: Chapter II, section 10 has no such result, but Chapter II,
section 5 does. The Theorem in the latter section is the correct formulation:
it is not the extension of the valuation, but the completion that one wants to
be unique. More or less coincidentally, the Proposition is correct as stated
(exercise!), but that statement is neither used (by anybody) nor proved (in the
book).

Page 45, line 5: for "=n" read "=n+1".

Page 52, part (3) of the first definition: for "K" read "V".

Page 54, line -5: replace roman "A" by italic "A" (twice).

Page 73, line 6: replace "vica" by "vice".

Page 75, line 1: replace "(19.9)" by "(19.10)".

Page 78, first line of display (A.19): replace "b_{ij}" by "b_{1j}".

Page 78, line -8 (display (A.24)): replace the third subscript "P" by "R".

Page 78, line -6: replace the third subscript "P" by "R".

Page 79, line -8: remove the three commas within the parentheses.

Page 98, the lower "delta" in the diagram should have a "hat" (the upper
one has one, though it is barely visible in my copy).

Page 123, last line before section 2.5: replace roman "C" by italic "C".

Page 140, line -2: replace "Prop. 2" by "Prop. 1". Also, the proof is
confused. One defines s'\alpha to be (\alpha,L'/K), and the fact that
s\alpha maps to s'_\alpha under the natural map G^{ab} -> (G/H)^{ab}
FOLLOWS from the equality of character values rather than playing a role
in the proof of that equality.

Page 141, first line after the first diagram: replace the last "K" by "K'".

Page 143, line -3: replace "Lubin" by "Lubin-".

Page 147, first line after Definition: put ")" at the end.

Page 150, proof of Proposition 1: (c) is not proved that way.

Page 150, line -10: for "left-and" read "left- and".

Page 151, line 13: replace the last "[a]" by "[b]".

Page 154, line 18: replace "r_\pi" by "r_\pi(\omega)".

Page 154, line 19: in my copy, there is the scrawled complaint "why is K_\pi
from sec. 3.6 equal to K_\pi from section 2.8?", and a three line additional
argument, which reads as follows: "Adopt the definition of K_\pi as in sec.3.6
(or Theorem 3(b)). Then r(\pi) is trivial on K_\pi (def. of r), and so is
\vartheta(\pi) by Cor. to Prop. 6. Also r(\pi) and \vartheta(\pi) are F on
K_{nr}. Hence r and \vartheta agree on \pi, hence on all of K^*. (Hence also
K_\pi=K\pi !)

Page 154, line 2 of section 3.8: replace "2.3" by "2.7".

Page 154, line -8: replace "I_K" by "I'_K".

Page 155, line -11: replace roman "G" by italic "G".

Page 156, line 3: replace "3.3" by "3.4".

Page 156, line 10: replace "\beta_j" by "\beta^j".

Page 157, line 9: replace "intertia" by "inertia".

Page 158, line -4: replace "|" by "/".

Page 168, line 5: replace roman "F" by italic "F".

Page 168, line -16: replace roman "C" by italic "C".

Page 170, line -18: replace "U^S an arbitrarily small" by "U^S contained in
an arbitrarily small" (because U^S is generally not open).

Page 175, line 2 after the diagram: for "N_{M/K}" read "N_{M/K}J_M".

Page 179, line 12: put ")" before the second "=".

Page 183, line 1: there is no "Proposition 2". Probably "Proposition 2.3" is
meant.

Page 183, display (7): replace the second "K" by "K^*".

Page 192, line -11: replace "infiinte" by "infinite".

Page 211, line -12 (counting the footnote as -1): for "does or does not"
read "does not or does". (This is what I scrawled, I did not verify it at the
present occasion!)

p. 126, line 3 of proof of Prop. 4: "for all sufficiently small open normal..."

p. 129, last line of 2nd paragraph: missing right parenthesis

p. 138, Application: seems K'/K should be assumed separable (and likewise the irreducible equation at the end should be separable).

p. 175, line above 6.3: in middle replace N_{M/K} with N_{M/K}(J_M).

p. 189, line -14: = 1, not = 0.

p. 190, line -12: source of \psi_p is Q*_p, not Q^{mc}_p.

p. 195, middle displayed formula for \beta_1(a), in next term should be \epsilon_2 between \beta_2 and (infl b).

p. 197, case r=-2, line 6: \widehat{H}^{-2}, not H^{-2}. (Also, the verification that the displayed map is actually inverse to the Artin map seems to merit a tiny bit of explanation (using the identification of artin map with cup product against fundamental class in Serre's treatment of the local case).

p. 198, line -10: not really typo, but ought to give a reference for this duality theorem, such as Thm. 6.6 in Ch, XII of Cartan-Eilenberg.

p. 200, displayed expression (15): lower right should be A (= A^{ab}), not E^{ab}, and the right vertical arrow is really composition of \psi_K with V:G_K^{ab} ----> G_L^{ab} = A.

p. 211, line -12: I think a_q being 0 or 1 cases should be swapped (see calculation of \zeta(f_{+/-}, +/-| |^s) on p. 317; the definition of \Phi(s,\chi) has also dropped some of the powers of \pi^{1/2} from those formulas, and this doesn't harm the statement of the functional equation but isn't quite "right" without them).

p. 212, line 14: -c_p, not +c_p.

p. 214 line 13 : should clarify that if \eta = 0 it means o(x).

p. 214, line 15: the definition of c is missing some symbols ([k:Q]), and the definitions of f and g should subtracting some stuff.

p. 215, line 6: displayed limit should be set equal to \ell.

p. 215, line -1: "absolute first degree relative to k".

p. 220, line 4: "we take as the local..."

p. 221, line -4: replace \frak{Q} with \frak{q}.

p. 222, line -8: \mu_0, not \mu, on left side.

p. 224, line -4: this is not really another way of arriving at a contradiction (distinct from citing linear independence of characters), since this argument basically is how one proves linear independence of characters over a field with characteristic not dividing the size of G.

p. 225, lines 1--3: this seems fishy, since the proposed procedure would involve inverting a gxg submatrix that depends on the choice of p. The Brauer argument below makes it all moot.